An equal area law for the van der Waals transition of holographic entanglement entropy

Transcription

1 UTTG TCC An equal area law for the van der Waals transition of holographic entanglement entropy arxiv: v2 [hep-th] 16 Aug 2015 Phuc H. Nguyen a,b a Theory Group, Department of Physics, University of Texas, Austin, TX 78712, USA b Texas Cosmology Center, University of Texas, Austin, TX 78712, USA phn229@physics.utexas.edu Abstract: The Anti-de Sitter-Reissner-Nordstrom (AdS-RN) black hole in the canonical ensemble undergoes a phase transition similar to the liquid-gas phase transition. i.e. the isocharges on the entropy-temperature plane develop an unstable branch when the charge is smaller than a critical value. It was later discovered that the isocharges on the entanglement entropy-temperature plane also exhibit the same van der Waalslike structure. In this paper, we present numerical results which sharpen this similarity between entanglement entropy and black hole entropy, by showing that both of these entropies obey Maxwell s equal area law. Moreover, we checked this for two disk-shaped entangling regions of different sizes, and the conclusion seems to be valid regardless of the region s size. We checked the equal area law for AdS-RN in 4 and 5 dimensions, so that the conclusion seems to hold for any dimension. Finally, we also checked that the equal area law holds for a similar, van der Waals-like transition of the dyonic black hole in AdS in a mixed ensemble (fixed electric potential and fixed magnetic charge), so that our conclusions seems to be true whenever the gravity background undergoes a van der Waals-like transition.

2 Contents 1 Introduction 1 2 Review of 4d AdS-RN in the canonical ensemble 3 3 Maxwell construction for entanglement entropy Brief review of holographic EE Maxwell s equal area law 6 4 Second example: dyonic AdS-RN The van der Waals transition of the mixed ensemble Maxwell s equal area law for entanglement entropy 10 5 Third example: 5d AdS-RN The van der Waals transition in the canonical ensemble Maxwell equal area law for entanglement entropy 12 6 Conclusion 13 7 Acknowledgements 15 1 Introduction Entanglement entropy (EE) appears to be a versatile tool that can be used to study a rich variety of physical phenomena. In particular, it can serve as a probe of the different phases of the theory [1, 2], ranging from the confining phase of large-n gauge theories [3] to topological phases in condensed matter systems [4], to tachyon condensation [5] and superconducting phase transitions [6]. Entanglement entropy has also emerged as a central component of the AdS/CFT correspondence. According to the Ryu-Takayanagi formula [7, 8], the entanglement entropy S A between a boundary region A and its complement is computed (in static backgrounds) in an elegant geometric fashion as the area of a minimal surface. The striking similarity between the Ryu-Takayanagi formula and the Bekenstein-Hawking formula for black hole entropy suggests some deep connection between entanglement entropy and black hole entropy. It has even been suggested that the origin of black hole entropy is entanglement entropy [9 11]. 1

3 Motivated by the themes above, in this paper we track entanglement entropy across two different phase transitions and demonstrate that entanglement entropy behaves in a strikingly similar way to black hole entropy. The first phase transition under study is the van der Waals-like transition of AdS-RN in 4 dimensions. This transition was first discovered in [12, 13]: the curves of constant charge on the temperature-entropy plane have an unstable portion when the charge is smaller than a critical value. Moreover, at critical charge, the unstable portion squeezes to an inflection point. It was subsequently pointed out in [14] that the same qualitative behavior is true if we study the isocharges in the entanglement entropy-temperature plane, and that the inflection point occurs at the same critical temperature as the black hole. These findings were then generalized to a wider class of supergravity backgrounds in [15], where it was found that, in all cases, the isocharges on the entanglement entropy-temperature plane mimick the qualitative behavior of the ones on the entropy-temperature plane. In this paper, we make these similarities more quantitative by studying Maxwell s equal area construction in the entanglement entropy-temperature plane. The equal area law in the context of black hole thermodynamics has generated some recent interest: this topic was studied in [16, 17] in the the context of AdS-RN, but also in [18, 19] in other contexts. We will show that the van der Waals behavior on the entanglement entropy - temperature plane also obeys Maxwells equal area construction, with the transition temperature obtained by minimizing the black hole free energy function. We present numerical results for two different entangling regions for AdS-RN in 4d: two boundary disks with different sizes, and show that the equal area construction holds regardless of the disks size. Moreover, we repeat the same procedure for AdS-RN in 5d, and also for a similar phase transition observed in the dyonic black hole in a mixed statistical ensemble (fixed electric potential and fixed magnetic charge). In all these cases, we showed that the equal area law holds. This indicates that the equal area law is a generic statement, and not specific to AdS-RN in a particular dimension. The rest of the paper is organized as follows: In Section 2, we review the phase structure of AdS-RN in 3+1 dimensions in the canonical ensemble and discuss Maxwell s equal area law in the entropy-temperature plane. In Section 3, we then turn to the numerical computation of holographic entanglement entropy, and present numerical evidence that the equal area construction also holds on the entanglement entropy-temperature plane. In Section 4, we repeat for the dyonic black hole in 3+1 dimensions and show that the equal area construction is also valid for this background. Next, in Section 5, we check that the equal area law also holds for AdS-RN in 4+1 dimensions. Finally, in Section 6, we summarize our main findings and discuss a few possible future work. 2

4 2 Review of 4d AdS-RN in the canonical ensemble In this section, we survey the phase structure of the 4-dimensional AdS-RN black hole in the fixed charge ensemble, leading up to the van der Waals behavior in the entropytemperature plane (i.e. there exists a family of first order transition ending with a second order one) and Maxwell construction 1. The Einstein-Maxwell action in 4 dimensions reads: I = 1 d 4 x g(r 2Λ F 2 ). (2.1) 16π The AdS-RN solution is given by: ds 2 = f(r)dt 2 + dr2 f(r) + r2 (dθ 2 + sin 2 θdφ 2 ), (2.2) f(r) = 1 2M r A = Q + Q2 r 2 ( 1 r + 1 r + r2 L 2, (2.3) ) dt. (2.4) where M is the mass, Q is the electric charge and L is the AdS-lengthscale. The additive constant in A t was chosen to be Q r + so that the norm of the vector potential A 2 is regular at the horizon. The black hole temperature and entropy are: T = 3r4 + + L 2 (r 2 + Q 2 ) 4πL 2 r 3 +, (2.5) S = πr 2 +, (2.6) where r + is the horizon (the largest root of f(r + ) = 0). From (2.5) and (2.6), we can easily eliminate the parameter r + to obtain the function T (S, Q): ( T (S, Q) = 1 ) 3 S π 4π L 2 π + π3/2 Q2. (2.7) S S 3/2 From the function T (S, Q) above, one can plot the isocharges on the T S plane. The plot is presented on the right panel of Figure 1. As can be seen from the plot, the curve is monotonic for sufficiently large Q. As Q decreases, the curve has an inflection point when Q reaches a threshold value Q c. One can solve for the position of the inflection point by: ( ) ( ) T 2 T = = 0. (2.8) S Q S 2 Q 1 The distinction between first order and second order here refers to the slope of the free energy plot versus the temperature. Upon a closer look, the nature of these phase transitions is more subtle: see for example [20] where the phase transition was studied using Ehrenfest equations. 3

5 We find the critical entropy S c, critical charge Q c and critical temperature T c to be: S c = π 6 L2, (2.9) Q c = L 6, (2.10) 2 1 T c = 3 πl. (2.11) Finally, when Q < Q c, the curve becomes oscillatory, and there is a small portion with negative heat capacity: ( ) S T 0. (2.12) T Q Like in the case of the liquid-gas transition, this portion is thermodynamically unstable, and should be replaced by an isotherm T = T according to Maxwell s prescription. The exact value of T can be obtained in two different (but equivalent) ways: by the equal area condition, or from the Helmholtz free energy. The first method, the equal area condition, states that T is the unique temperature which divides the oscillatory part of the curve T (S) into two regions with equal area. In the remainder of this section, we will find T from the second method, i.e. using the Helmholtz free energy, and check numerically that it is equivalent to the first. The Helmholtz free energy can be found from the on-shell action 2 : F = 1 4L 2 ) (L 2 r + r Q2 L 2. (2.13) We present in the left panel of Figure 1 the plot of F versus T. For Q < Q c, we observe the swallowtail behavior familiar from catastrophe theory, and the transition temperature T is the horizontal coordinate of the junction between the two stable branches. Numerically, we found T We can now check that Maxwell s equal area law holds, by checking the equivalent statement: T (S 3 S 1 ) = S3 r + S 1 T (S, Q)dS, (2.14) where S 1 and S 3 are the smallest and largest roots of the equation T = T (S, Q). Numerically, we found that both sides of (2.14) evaluate to , thus confirming the validity of Maxwell s construction. 2 The Helmholtz free energy of AdS-RN is usually measured with respect to an extremal background with the same electric charge [12]. For our purposes, however, this background subtraction only shifts the plot of F versus T in the vertical direction and does not affect the transition temperature T. 4

6 F T T S Figure 1. Left panel: Plot of the free energy versus the temperature, with Q = 1.5 (green), Q = Q c = 10 6 (red) and Q = 1.8 (orange). Right panel: Plot of isocharges on the T S plane with the same 3 values of the charge: Q = 1.5 (green), Q = Q c = 10 6 (red) and Q = 1.8 (orange). The transition temperature for the green curve is T = Maxwell construction for entanglement entropy In this section, we investigate the phase structure of the EE-temperature plane instead of the entropy-temperature plane. As discussed in the introduction, the work of [14] demonstrates that, at least for AdS-RN in 4d, the isocharges on the EE-temperature plane mimick the qualitative behavior of the entropy-temperature plane. However, that paper did not make the connection with Maxwell s construction. In this section, we will show that, indeed, entanglement entropy also respects the Maxwell construction, thereby strengthening the conclusion of [14]. But first, let us briefly review a few field theory generalities about EE. 3.1 Brief review of holographic EE Suppose we have a quantum field theory described by a density matrix ρ, and let A be some region of a Cauchy surface of spacetime. The entanglement entropy between A and its complement A c is defined to be: S A = Tr A (ρ A log ρ A ), (3.1) where ρ A is the reduced density matrix of A: ρ A = Tr A c(ρ). As mentioned in the introduction, entanglement entropy is computed holographically by the Ryu-Takayanagi 5

7 recipe 3 : S A = Area(Γ A) 4G N, (3.2) where Γ A is a codimension-2 minimal surface with boundary condition Γ A = A, and G N is the gravitational Newton s constant. A few remarks are in order concerning EE at finite temperature and in finite volume. At nonzero temperature, entanglement entropy no longer has the nice properties that it does at zero temperature, for example the area law (see, for example, [21]). This is due to the fact that, at finite temperature, entanglement entropy is contaminated by a thermal component which scales as the volume of the entangling region rather than its area. For this reason, we will choose a small entangling region in order to filter out the thermal part. Moreover, when the bulk is topologically nontrivial (as is the case for AdS-RN with spherical horizon), the Ryu-Takayanagi formula is refined by an additional topological constraint: only surfaces which are homological to the entangling region on the boundary are considered in the minimization problem [22]. This constraint, which ensures that the Araki-Lieb inequality is satisfied, implies that, for sufficiently large entangling region, the minimal surface is disconnected and includes the horizon itself as a connected component. It might appear curious that the homology constraint means the entanglement entropy of A is not equal to that of the complement of A, but this is expected when the system is in a mixed state. By choosing a small entangling region, we will also avoid having to deal with this phase transition between connected and disconnected minimal surfaces Maxwell s equal area law We will take the region A to be a spherical cap on the boundary delimited by θ θ 0. From the remarks in the previous subsection, we will pick θ 0 to be small. We will show that our findings are valid regardless of the value of θ 0, so we will consider two different values of θ 0 : 0.15 and 0.1. The minimal surface can be parametrized by the function r(θ), which is independent of the coordinate φ by rotational symmetry. The area functional is: θ0 (r A = 2π r sin θ ) 2 0 f(r) + r2 dθ, (3.3) where r dr. The function r(θ) is the obtained by solving the Euler-Lagrange equation: dθ L r = d dθ ( ) L, (3.4) r 3 The Ryu-Takayanagi formula only applies to static backgrounds and when the bulk theory is Einstein gravity. 4 Similar phase transitions are also observed in infinite volume, when the entangling region is composed of multiple strips [23]. 6

8 with the boundary conditions r(θ 0 ) and r (0) = 0 (i.e. the minimal surface is regular at the center θ = 0). Also, since EE is UV-divergent, it has to be regularized. We will do it by subtracting the area of the minimal surface in pure AdS whose boundary is also θ = θ 0. In other words, we first integrate the area functional to some cutoff θ c θ 0. Then we set M = Q = 0 to obtain AdS in global coordinates: ds 2 = (1 + r2 L 2 )dt2 + dr2 + r 2 dω r2 2. (3.5) L 2 The minimal surface which goes to θ = θ 0 on the boundary is given by ( ( ) ) 2 1/2 cos θ r AdS (θ) = L 1. (3.6) cos θ 0 We can easily integrate the area functional of this minimal surface up to θ c. Then we subtract this quantity from the black hole one to obtain the renormalized entanglement entropy, which we will denote by S A. For the numerical computation, we choose θ c to be when θ 0 = 0.15, and θ c to be when θ 0 = 0.1. In Figure 2, we plot 3 isocharges on the S A T plane, using the same 3 values of the electric charge as in the previous section. As can be seen on this plot, the van der Waals behavior noted in [14] is observed. We have also drawn the transition isotherm in dashed green taken from the free energy function. If the Maxwell construction is to hold, it would amount to the statement: where S (1) A T. T ( S (3) A S(1) (3) S A ) = A S (1) A T ( S A, Q)d S A, (3.7) and S(3) A are the smallest and largest roots of the equation T ( S A, Q) = Numerically, we find that, for the case θ 0 = 0.15, the left-hand side of (3.7) evaluates to , while the right-hand side evaluates to For the case θ 0 = 0.1, the left-hand side of (3.7) evaluates to and the right-hand side evaluates to Therefore, for both values of θ 0, we find excellent agreement between the two sides of (3.7), ans we can safely conclude that Maxwell construction is valid for entanglement entropy (independently of the size of the entangling region). 4 Second example: dyonic AdS-RN In this section, we study another example of Maxwell s equal area law for entanglement entropy: the dyonic AdS-RN solution in 4d, which describes a black hole in AdS carrying both an electric charge and a magnetic charge. The dyonic AdS-RN black hole is also a 7

9 T ΔS A T ΔS A Figure 2. Plot of isocharges on the T S A plane with Q = 1.5 (green), Q = Q c = 10 6 (red) and Q = 1.8 (orange) using θ 0 = 0.15 (top) and θ 0 = 0.1 (bottom). In both plots, the transition isotherm is obtained from the free energy (left panel of Figure 1). For the green curves, we also show the data points which were used to create the interpolation. solution to the Einstein-Maxwell action (2.1). The line element together with the gauge field are given by [24 26]: ds 2 = f(r)dt 2 + dr2 f(r) + r2 (dθ 2 + sin 2 θdφ 2 ), (4.1) ( 1 A = Q 1 ) dt + P cos θdφ, r + r (4.2) 8

10 with Here P is the magnetic charge. f(r) = 1 2M r + Q2 + P 2 r 2 + r2 L 2. (4.3) 4.1 The van der Waals transition of the mixed ensemble As for the AdS-RN solution, the phase structure depends on which statistical ensemble one chooses. One can, for example, study the ensemble with both the electric and magnetic charges fixed. This is however not very interesting, since the results in this case can be trivially obtained from the results in section 2 with the replacement Q 2 Q 2 +P 2. Thus, we will instead work in the fixed Φ, fixed P ensemble. It was observed in [26] that the phase structure in this mixed ensemble exhibits the van der Waals transition. The asymptotic value of the electric potential Φ is related to the electric charge Q by: The temperature and entropy are given by: T = 1 4πr + Φ = Q r +. (4.4) ( 1 + 3r2 + L 2 Φ2 P 2 r 2 + ), (4.5) S = πr 2 +. (4.6) From which we find the function T (S, Φ, P ). Also, since the electric charge is now allowed to vary, we should compute the Gibbs potential: G = M T S ΦQ, (4.7) instead of the Helmholtz free energy. Computation gives: G = 3P 2 + r ( ) + 1 Φ 2 r2 +. (4.8) 4r + 4 L 2 If we keep Φ fixed, and plot the curves of constant P on the S T plane, we can observe a van der Waals-like transition when P is smaller than a critical value P c. The plot of these curves is presented on the right panel of Figure 3. The critical magnetic charge P c, entropy S c and temperature T c can be found to be: P c = L 6 (1 Φ2 ), (4.9) S c = L2 6 π(1 Φ2 ), (4.10) 2 1 T c = 1 Φ2. (4.11) 3 Lπ We also plot the Gibbs potential on the left panel of Figure 3. From this plot, we obtained the transition temperature T = for the green curve, then checked numerically that T is indeed the temperature which obeys Maxwell s equal area law: numerical integration of both sides of (2.14) yields a value of

11 0.80 G T T S Figure 3. Left panel: Plot of the Gibbs potential versus the temperature, with P = 0.95 (green), P = P c = (red) and P = 1.3 (orange), and with Φ = 0.6 for all three curves. Right panel: Plot of isocharges on the T S plane with the same 3 values of P and the same value of Φ. The transition temperature for the green curve is T = Maxwell s equal area law for entanglement entropy Next, we turn to the computation of entanglement entropy. Like in section 3.2, we consider the disk-shaped region θ θ 0 with two values of θ 0 : 0.15 and 0.1. The plots of temperature versus entanglement entropy are presented in Figure 4. In these plots we have chosen Φ = 0.6 and plot curves of constant magnetic charge for 3 different values of P. As for the AdS-RN case, we will check the Maxwell s equal area law by computing both sides of the criterion (3.7). For θ 0 = 0.1, we find that the left-hand side evaluates to and the right-hand side evaluates to For θ 0 = 0.15, we find that the left-hand side evaluates to and the right-hand side evaluates to Therefore, like in the AdS-RN case, we can safely conclude that the equal area law works for the dyonic solution, again independently of the choice of θ 0. 5 Third example: 5d AdS-RN In this section, we will show that the equal area law for entanglement entropy also works for AdS-RN in 4+1 dimensions. 10

12 T ΔS A T ΔS A Figure 4. Plot of isocharges on the T S A plane for the dyonic black hole in the mixed ensemble. The colors correspond to P = 0.95 (green), P = P c = (red) and P = 1.3 (orange). We also use θ 0 = 0.15 (top) and θ 0 = 0.1 (bottom), and Φ = 0.6 for both panels. In both plots, the dashed isotherm is obtained from the Gibbs potential (left panel of Figure 3). For the green curves, we also show the data points which were used to create the interpolation. 5.1 The van der Waals transition in the canonical ensemble The AdS-RN solution in arbitrary dimension is given in [12]. For 4+1 dimension, the metric is: ds 2 = f(r)dt 2 + dr2 f(r) + r2 (dψ 2 + sin 2 ψ(dθ 2 + sin 2 θdφ 2 )) (5.1) 11

13 where (ψ, θ, φ) are hyperspherical coordinates on the 3-sphere, with ψ and θ ranging from 0 to π, and φ ranging from 0 to 2π, and f(r) = 1 8M 3πr 2 + 4Q2 3π 2 r 4 + r2 L 2 (5.2) The black hole temperature and entropy are given in terms of the horizon r + by: T = r + πl πr + 2Q2 3π 3 r 5 + (5.3) S = π2 2 r3 + (5.4) From which we obtain the function T (S, Q). Following the same steps as the 4-dimensional case, we plot the isocharges on the T S plane on the right panel of Figure 5. We find the inflection point to be at: T c = 4 3 5πL The free energy in the fixed charge ensemble is now: F = π 8L 2 π3 S c = 6 3 L3 (5.5) Q c = π 6 5 L2 (5.6) ) (L 2 r 2+ r Q2 L 2 3π 2 r 2 + (5.7) (5.8) We plot the free energy versus the temperature on the left panel of Figure 5. The phase structure again presents a van der Waals transition, like in 4d, but we only present a curve with Q < Q c. The transition temperature was found to be T = We can now check numerically the equal area law: evaluating the left-hand side of 2.14 numerically yields , and the right-hand side yields Maxwell equal area law for entanglement entropy Next, we turn to computing the entanglement entropy of a spherical region. The 5d analog of the disk-shaped region θ θ 0 = 0.15 in 4d is the spherical region ψ ψ 0 for some constant ψ 0 [0, π], which we will choose to be ψ 0 = The minimal surface can be parametrized by r(ψ), where this function does not depend on θ or φ by rotational symmetry. The area functional is given by: ψ0 A = 4π r 2 sin 2 ψ r 2 + (r ) 2 dψ (5.9) 0 f(r) 12

14 F T T S Figure 5. Left panel: Plot of the free energy versus the temperature for AdS-RN in 4+1 dimensions, with L = 3 and Q = 1. Right panel: Plot of isocharges on the T S plane also with L = 3 and Q = 1. The transition temperature for the green curve is T = Proceeding as for the 4d case, we will solve this equation numerically with the boundary condition that the minimal surface is regular at the center. To regularize entanglement entropy, we again subtract the pure AdS minimal surface which goes to the same ψ 0 at the boundary. Such a surface is analytically given by: ( ( ) ) 2 1/2 cos ψ r AdS (ψ) = L 1 (5.10) cos ψ 0 Like in the 3+1 dimensional case, we choose the cutoff value of ψ to be ψ cutoff = We present in Figure 6 an isocharge (with charge Q = 1 smaller than the critical charge, and L = 3) which we use to check the equal area law. The left-hand side of (3.7) evaluates to , while the right-hand side evaluates to Thus, once again, it seems that the equal area law is valid. 6 Conclusion The laws of black hole thermodynamics have provided us with a robust understanding of black holes as thermodynamical systems, though the nature of the microscopic constituents of these systems remains by and large mysterious. The gauge-gravity duality seems to suggest, on very general ground, that gravity could emerge from the dynamics of a strongly coupled large-n quantum field theory. In view of this, it is an important 13

15 T ΔS A Figure 6. Plot of a subcritical isocharge on the T S A plane for the 4+1 dimensional AdS-RN solution, with L = 3, Q = 1 and θ 0 = The transition isotherm is obtained from the free energy (left panel of Figure 5). We also show the data points which were used to create the interpolation. thing to do to see whether observables in such a quantum field theory can mimick the behavior of gravitational systems. In this paper, we have presented compelling numerical evidence that the van der Waals-like phase structure observed in charged AdS black holes is also observed on the level of entanglement entropy. Specifically, we have improved upon previous studies such as [14] by showing that Maxwell s equal area law is valid even for entanglement entropy! Moreover, the Maxwell s construction for entanglement entropy seems to be generically valid: it does not depend on the specific size of the entangling region, the particular gravity background (as long as it has a van der Waals-like phase transition), or the dimension. That said, we have only considered spherically symmetric entangling regions, and future work to generalize the conclusions in this paper to entangling regions of other shape could yield interesting insights. Finally, we note that the van der Waals transition of AdS-RN has been observed in the context of the extended black hole thermodynamics, where the cosmological constant is identified as the pressure variable and its thermodynamic conjugate as the volume variable [27]. In this framework, one can analyze the isotherms on the P V plane, and they turn out to be remarkably similar to the van der Waals gas. One can wonder about the relationship between the van der Waals transition in the P V plane and the 14

16 one in the T S plane. It was pointed in [16] that the two are related to each other by a duality similar to T-duality of string theory. Moreover, it was noted by [19] that Maxwell s construction in the P V plane can be surprisingly subtle (the construction does not work if the volume is replaced by the specific volume, unlike the usual van der Waals gas). Finally, we note that the holographic interpretation of the P V plane is not very well-understood. However, the van der Waals transition on the P-V plane seems to be connected with the renormalization group flow (see [15, 28]). Finally, another interesting direction of investigation is to generalize our findings for the dyonic AdS-RN black hole to other, more complicated configurations with magnetic field. For instance, there has been lots of recent efforts to construct magnetic stars in AdS (see, among others, [29 31]) which are the gravitational duals to condensed matter systems. In particular, entanglement entropy has been studied in such a background in [32]. 7 Acknowledgements It is a pleasure to thank Elena Caceres, Willy Fischler, Richard Matzner and Juan Pedraza for discussions and comments on the manuscript. This research was supported by the National Science Foundation under Grant PHY References [1] P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406, P06002 (2004) [hep-th/ ]. [2] P. Calabrese and J. L. Cardy, Entanglement entropy and quantum field theory: A Non-technical introduction, Int. J. Quant. Inf. 4, 429 (2006) [quant-ph/ ]. [3] I. R. Klebanov, D. Kutasov and A. Murugan, Entanglement as a probe of confinement, Nucl. Phys. B 796, 274 (2008) [arxiv: [hep-th]]. [4] A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96, (2006) [hep-th/ ]. [5] T. Nishioka and T. Takayanagi, AdS Bubbles, Entropy and Closed String Tachyons, JHEP 0701, 090 (2007) [hep-th/ ]. [6] T. Albash and C. V. Johnson, Holographic Studies of Entanglement Entropy in Superconductors, JHEP 1205, 079 (2012) [arxiv: [hep-th]]. [7] S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96, (2006) [hep-th/ ]. [8] T. Nishioka, S. Ryu and T. Takayanagi, Holographic Entanglement Entropy: An Overview, J. Phys. A 42, (2009) [arxiv: [hep-th]]. [9] M. Srednicki, Entropy and area, Phys. Rev. Lett. 71, 666 (1993) [hep-th/ ]. 15

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